Theorems · Theorem · functional analysis
cfcHom_map_spectrum
∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
[inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
[inst_7 : StarRing A] [inst_8 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] {a : A} (ha : p a)
(f : C(↑(spectrum R a), R)), spectrum R ((cfcHom ha) f) = Set.range ⇑fThe spectral mapping theorem for the continuous functional calculus.
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- Set.Elemstatement and proof · cited by 7,166
- Set.rangestatement · cited by 4,705
- ContinuousMapstatement and proof · cited by 2,491
- StarRingstatement and proof · cited by 1,686
- MetricSpacestatement and proof · cited by 1,684
- ContinuousStarstatement and proof · cited by 543
Cited by7
Results whose statement or proof uses this declaration.
- cfc_map_spectrumproof · cited by 11
- cfc_compproof · cited by 5
- cfcHom_nonneg_iffproof · cited by 3
- SpectrumRestricts.cfcproof · cited by 2
- cfcHom_isStrictlyPositive_iffproof · cited by 1
- spec_cfcₙAuxproof · cited by 1
- cfcₙHom_of_cfcHom_map_quasispectrumproof · cited by 0